Linear independence for C_⁽¹⁾ by using C₂⁽¹⁾

In this note we prove linear independence of the combinatorial spanning set for standard C_⁽¹⁾-module L(kΛ₀) by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace W(kΛ₀) of C₂⁽¹⁾-module L(kΛ₀). It should be noted that the proof of linear independence for the basis of W(kΛ₀) is obtained by using simple currents and intertwining operators in the vertex operator algebra L(kΛ₀).